[Proof of Proposition REF ]
Let us prove the second result. Let \(\delta >0\) . Bennett's inequality ([1]}, Theorem 3 in [2]}) implies that with probability at least \(1-\delta \)
\(\mathbb {E}_{\mathbb {P}}(\ell (x,\xi )) - \mathbb {E}_{\hat{\mathbb {P}}_T}(\ell (x,\xi ))\ge -\sqrt{\frac{2\ln 1/\delta }{T}\mathrm {Var}_{\mathbb {P}}(\ell (x,\xi ))}-\frac{K\ln 1/\delta }{3T}.\) 
